Question

Given below are two statements:
Statement (I): The dimensions of Planck's constant and angular momentum are same.
Statement (II): In Bohr's model, electrons revolve around the nucleus only in those orbits for which angular momentum is an integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Statement I is incorrect but Statement II is correct
• C. Statement I is correct but Statement II is incorrect
• D. Both Statement I and Statement II are incorrect

Hint:

Always remember Bohr’s quantization rule: \[ mvr = n\frac{h}{2\pi} \] Angular momentum is quantized in units of \( \frac{h}{2\pi} \), not \(h\).

Answer 1

The Correct Option is C

Analysis of Statements:

We evaluate the statements by comparing physical units and applying the definition of Bohr's quantization.

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Step 1: Check Unit Equivalence (Statement I)

Instead of deriving dimensional mass/length/time formulas, we can compare their standard SI units.

For Planck's constant ($h$), using the photon energy equation $E = h\nu$:
$h = \frac{E}{\nu} \Rightarrow \frac{\text{Joule}}{\text{s}^{-1}} = \text{Joule} \cdot \text{second}$

For Angular momentum ($L$), using the formula $L = r \times p$:
$L = \text{meter} \times (\text{kg} \cdot \text{m/s}) = \text{kg} \cdot \text{m}^2/\text{s}$
Since $1 \text{ Joule} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2$, we can rewrite the unit of $L$ as:
$L = (\text{kg} \cdot \text{m}^2/\text{s}^2) \cdot \text{s} = \text{Joule} \cdot \text{second}$

Since both quantities share the unit $\text{J} \cdot \text{s}$, they have identical dimensions.
Therefore, Statement (I) is correct.

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Step 2: Check Quantization Condition (Statement II)

Statement (II) claims angular momentum is an integral multiple of $h$.

Recall the fundamental postulate of the Bohr model:
$L = n\hbar$

Where $\hbar$ (h-bar) is the reduced Planck's constant:
$\hbar = \frac{h}{2\pi}$

Therefore, the correct relation is:
$L = n \left( \frac{h}{2\pi} \right)$

Because of the factor $\frac{1}{2\pi}$, angular momentum is not an integral multiple of $h$ directly.
Therefore, Statement (II) is incorrect.

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Final Conclusion:

Statement (I) is correct and Statement (II) is incorrect.